Reading this wiki article, it seems to me that 'with the aid of' in the following sentence is a prepositional object. I suspect I may be wrong and would like a simple explanation about the grammatical functions of 'with the aid of' in the following sentence:
With the aid of a physiotherapist, everything turned out fine.
In the terminology of CGEL, with the aid of a physiotherapist is an adjunct (meaning, one can remove it and the sentence is still grammatical even when all the words in the remainder are taken in the vary same meanings they had before the removal). More precisely, it is an adjunct of means (as in, 'by what means' was something done). Other sources might call it an adverbial of manner. However it is called, it is realized by a preposition phrase (PP). The head of this PP is the preposition with, and the complement of this preposition is the noun phrase (NP) the aid of a physiotherapist. The whole PP is fronted, though it could also be placed at the end, as in Everything turned out fine with the aid of a physiotherapist.
Note that with the aid of is actually not a syntactical constituent of the sentence. The constituent structure is, rather, this: [With [the [aid [of [a physiotherapist] ] ]. As I said above, the whole thing is a PP whose head is with and whose complement is the NP the aid of a physiotherapist. That NP, in turn, is composed of the determiner the and the nominal aid of a physiotherapist. The head of the nominal is the noun aid. The nominal also has a post-head complement, which is the PP of a physiotherapist (it is a complement rather than a modifier because the head noun licences the particular preposition; aid here cannot be followed by just any PP, but only by PPs headed by a very restricted set of prepositions, such as of. For example, on would not work at all, regardless of what followed it). Finally, this final PP has a complement, the NP a physiotherapist, which consisits of a determiner, the indefinite article, and the nominal, which is the noun physiotherapist.
